A Comprehensive System for Modeling Variation in Mechanical Assemblies. ADCATS Report No. 93-2

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1 A Comprehensive System for Modeling Variation in Mechanical Assemblies ADCATS Report No Angela Trego, graduate student Department of Mechanical Engineering Brigham Young University MS Thesis sponsored by ADCATS April 1993 ABSTRACT Tolerance analysis allows the designer to quantitatively estimate the affects of variation on design requirements in the early design phases. Tolerances play a significant role in the development and cost of manufactured products. By creating assemblies which perform properly, are cost efficient and readily manufacturable, engineers can assist in producing high quality, marketable products. Research presented in this thesis focuses on developing a comprehensive 2-D system for modeling manufacturing variations. Previous studies included creating an engineering model based on part, datums, kinematic joints, vector loops, and form variations. Contributions from this thesis include: 1) A comprehensive system for defining assembly tolerance specifications. Just as feature controls are applied to individual parts, a similar system of assembly tolerance specifications may be applied to assemblies of parts. 2) A generalized model for manual process variations due to fastener clearances or other non-deterministic variations. 3) An algorithm for automatic open loop generation. Open vector loops are often necessary to represent design constraints on an assembly. 4) A method for detecting degree of freedom redundancy.

2 CHAPTER 1: INTRODUCTION Tolerances play a significant role in the development and cost of manufactured products. Tolerances are required on mechanical assemblies to ensure proper assembly and function, but at the same time tight tolerances drive up the cost of manufacturing (see figure 1.1). A product s manufacturability is important in the design process. If the product cannot be manufactured profitably, the designer has failed. By creating assemblies which are useful, cost efficient and readily manufacturable, engineers can assist in producing high quality, marketable products. Cost Minimum Total Cost Cost due to Tightening Tolerances Cost due to Rejected Assemblies Optimal Tolerance Selection Total Cost Tight Tolerances Loose Figure 1.1. Cost versus Tolerances. All manufactured parts exhibit variation which may accumulate in an assembly. As a result of the variation or tolerance stack-up the assembly may not function properly. Tolerance analysis allows the designer to quantitatively estimate the affects of variation on design requirements. It ties the component tolerance requirements, which manufacturing systems people must meet, to the assembly tolerance requirements, which result from engineering design requirements. Applying tolerance analysis in the early design phases, allows the designer to save valuable time and money by decreasing rework costs. As shown in figure 1.2, rework cost increases dramatically as a design progresses from concept to sales. Profit loss due to design changes therefore increases at later phases of the product development processes. It is desirable to do a better job of design in the early stages to avoid costly changes later. New programs, such as concurrent or simultaneous

3 Introduction 2 engineering, design for manufacture, and design for assembly are attempts to formalize the process of considering the manufacturing consequences of design decisions. $500,000 Cost per Design Change $50,000 $5,000 $500 $50 Conceptual Model Detailed Design Prototyping & Testing Production Sales & Marketing Figure 1.2. Cost of design changes. The next section will describe a new quantitative tool for integrating manufacturing considerations into the engineering design process. CAD Tolerance Modeling Engineers are beginning to utilize CAD/CAM for more than just drawings, but also for applying tolerance modeling and analysis in the early design stages. A CAD database stores the geometric information required for tolerance analysis. By creating an interface with the CAD system, information may be directly accessed from the drawings, instead of tediously determined by hand. Engineering models define a set of relationships which describe how different assembly parts interact due to manufacturing variations. These relationships are in the form of vector loops joined by kinematic joints placed at the contact points between pairs of mating parts. These engineering models of assemblies require complex equations. CAD modeling systems allow the equations to be created graphically and stored as part of the CAD model. CAD/CAM drawings then allow the designer to change tolerances with a stroke of a key.

4 Introduction 3 Menu Interface Tolerance Modeler Model File Tolerance Analysis Package CAD Sytem CAD Database Figure 1.3. CAD tolerance modeling system with analysis package. Figure 1.3 illustrates a new tolerance modeling system which is integrated with a CAD system. A menu-driven, user-interface commands the modeler and appends the assembly tolerance model to the CAD database just as the CAD system itself accesses geometry from its database. Once the model has been created, a neutral file, or file containing essential information needed to perform tolerance analysis is generated. This model file may then be sent to the analysis package. The assembly tolerance analysis package predicts the magnitude of the variations which occur at assembly time. Results are then passed back to the CAD system along with tolerance modifications for incorporation into the drawings. Research Objectives The objectives of this thesis are to 1) create a more complete assembly modeling system which is capable of representing a wider range of assembly problems, and 2) develop algorithms for automating more assembly modeling tasks in order to decrease the modeling expertise required by design engineers and to reduce modeling errors. Method A comprehensive 2-D system will be developed for defining assembly tolerance specifications, modeled after the ANSI Y14.5 system for feature controls on single parts. This work is being carried out jointly with David Carr who is currently working on methods for solving open loop equations for each new design specification [Carr, 1993].

5 Introduction 4 A new modeling element called manual process variations will also be defined. The data structure and neutral file will be updated to allow for the new modeling elements. In addition, an improved system for testing degree of freedom (DOF) redundancy will be developed. Algorithms for automatic open loop generation will be studied, complimenting existing algorithms for closed loops. Design procedures will be examined and a more intuitive system for adding these entities to a tolerance model will be devised. Finally, the new modeling techniques will be implemented into AutoCAD using the programming language AutoLISP. Overview The following chapter discusses research related to computer aided tolerance modeling and analysis. Chapter three introduces assembly tolerances or design specifications. Automation of kinematic chain or open vector loop generation is discussed in chapter four. Analysis and user enhancements are presented in chapter five. Chapter six presents example assemblies to be modeled and analyzed. Conclusions and recommendations for future research are in chapter seven.

6 CHAPTER 2: BACKGROUND Recently, research on computer-aided tolerance modeling using CAD systems has grown with many new applications and developments. This chapter will focus on first, general tolerance modeling requirements and secondly, the existing CAD tolerance modeling system. Tolerance Modeling Requirements With most assemblies stored in CAD systems, precise geometric data is available for engineering applications. Although this provides a great deal of information, it is not sufficient for assembly tolerance analysis. Srikanth has pointed out that graphical representation of individual part interfaces may be provided by CAD databases, but CAD applications end there [Srikanth et al., 1990; Shah et al., 1990]. The CAD systems need to provide more than just graphical input of geometry, rather they need to provide kinematic relationships between mating parts and assembly constraints. Ranyak and Fridshal [Ranyak, 1988] created a tolerance analysis method on a featurebased solid modeler. They created a hierarchical feature model for the specific purpose of defining tolerances on the feature. Since a feature is to a part as a part is to an assembly, this method was limited to tolerance specifications for single parts only. CAD systems needs to provide information on tolerance limits applied to the size, form, orientation and location of each feature of a component, as well as the assembly tolerance limits set by engineering design requirements. Srikanth and Turner took the feature based method one step further. They discuss an assembly hierarchy which includes mating features between individual parts [Srikanth et al., 1990]. The kinematic relationships are then combined with geometric information creating an entire mechanical assembly model.

7 Background 6 Larsen [Larsen 1991] noted that assembly tolerance analysis models need to accurately account for: Relative rigid body translation and rotation between components Propagation of size variations Propagation of form and feature variations Propagation of kinematic adjustments Tolerance accumulation Rigid body motion is the resultant motion within an assembly due to a single part's variation. Every feature on a part deviates from the ideal design. Form and feature variations also must be modeled just as size variations need to be modeled. Finally, kinematic adjustments, or adjustments at assembly time due to varied part dimensions, are a crucial part in accurately modeling assembly variations. According to Shah and Miller, a tolerance model, besides being geometrically associated with entities in the CAD database, must support geometric tolerances (ANSI Y14.5 specifications), datum reference frames and datum precedence, provide default tolerances and graphical feedback [Shah, Miller 1990; Larsen 1991]. Faux [Faux, 1981] also supports the need for an assembly data structure which includes mating features and the nature of fit between assembly components. A CAD tolerance modeler, such as figure 1.1, which includes these qualities could provide the essential graphical interface which allows both geometric and kinematic information to be analyzed. Many steps have been taken to model all assembly variations correctly using a CAD database system such as figure 1.1 [Larsen, 1991; Marler, 1988; Robison, 1989; Ward, 1992]. For an assembly tolerance modeler to be complete it must also have the capability of providing parameters to the governing equations of an assembly. It is desirable to automate this process as much a possible. Wang introduced a method of generating kinematic chains automatically with the application of network graph theory [Wang, 1990]. This theory has been developed for a limited type of assembly models [Robison, 1989; Simmons, 1990; Larsen, 1991]. Now that the general requirements for an assembly tolerance modeler have been defined, an in-depth look at the existing CAD modeling system which is based on this approach will be discussed. This existing system forms the starting basis for the research of this thesis.

8 Background 7 Modeling Assemblies The existing prototype modeling package, AutoCATS, uses a hierarchical structure as suggested by both Ranyak and Srikanth [Ranyak, 1988; Srikanth et al., 1990]. CATS.BYU, Computer-Aided Tolerance Specification, is a system for assembly tolerance analysis which has been developed through research efforts at Brigham Young University since The software allows the designer to create 2-D assembly models using a graphical preprocessor called AutoCATS, and use these models for predicting design consequences of manufacturing variations. The assembly drawing is first created in a CAD system. Using a step-by-step procedure, a model is created using parts, feature datums, kinematic joints, form tolerances, and vector loops (figure 2.1). Datum reference frames distinguish each part of an assembly. Kinematic joints relate contact points between individual parts. They are located relative to each part by datum paths, or vector paths. Form tolerances may also be applied at each interface. Vector loops are then created which join relevant dimensions into chains describing resultant assembly dimensions and related features. A neutral file is then generated which may be sent to an analysis software package which generates the governing equations. The equations permit the analysis of tolerance accumulation and prediction of assembly reject rates due to failure to meet assembly tolerance limits.

9 Background 8 Procedure for Creating AutoCATS Models for Tolerance Analysis of Assemblies USER INPUT Assembly drawing AUTOCATS Load AutoCAD assembly drawing Part name, location and type Name and define for each part Joint location, type and orientation Create kinematic joints Degrees of freedom check * Automatic loop generation * Specify tolerances Identification of kinematic variables Specify tolerances and type Add feature control tolerances (ANSI Y14.5) Specify type, location and tolerance Add design specifications * Generation of AutoCATS model file Figure 2.1. Modeling procedure for AutoCATS. * New modeling steps by author PARTS In order to distinguish between individual parts of an assembly and to permit datumreferenced dimensioning, a datum reference frame () is created on each part. A is a local coordinate system for each part. During production, all part features are ultimately referenced to these specified datum planes used to fixture the part. Selection of the location for the origin determines to a great extent which part dimensions will contribute to assembly variations. Figure 2.2 shows a three part assembly with a defined on each part.

10 Background 9 Cylinder Block Ground Figure 2.2. Assembly with Parts and s defined. JOINTS The points of contact between mating assembly parts are called joints. Kim and Lee first developed the system of deriving an assembly model only from mating conditions [Kim et al., 1989]. A joint defines a kinematic pair which constrains the relative motion between two mating parts. For example, a block on a plane is constrained to slide parallel to the plane. There are six joint types which model a wide variety of 2-D assembly conditions. They are shown below in figure 2.3 with their associated degrees of freedom. Each joint type reduces the degrees of freedom of an assembly by constraining motion. Rigid (no motion) Revolute (1) Cylindrical (1) Planar (1) Edge Slider (2) Cylindrical Slider (2) Figure D joint types with associated degrees of freedom (DOF). Each joint must be located relative to the of both parts connected by the joint. The chain of vectors which locates the joint from a datum reference frame () is called a datum path. Each vector in a datum path must be either a controlled component dimension, for which the designer may specify a tolerance, or a kinematic assembly dimension, which adjusts at assembly time. Kinematic dimensions are determined by a

11 Background 10 chain of component dimensions, hence the tolerances are the result of a tolerance stack-up of component tolerances. LOOPS The relationships representing the assembly, both geometrically and kinematically, are obtained by creating a set of vector loops which connect contact joints in the assembly. The loops may be either open or closed: Closed loops start and end at the same location and represent kinematic constraints on the assembly. For example, a kinematic constraint may state that all parts in the assembly must maintain contact in order for the tolerance model to be valid [Larsen, 1991]. Open loops are used to determine assembly resultants of interest such as a clearance, orientation or position. For example, a fan blade must maintain a certain clearance in an assembly to operate properly. In the previous system, closed loops were generated either manually or automatically using a generalized modeling approach [Larsen, 1991]. Larsen's algorithm was limited to the generation of closed loops only and did not allow for zero length vectors in the loop. Once a complete loop is formed from joints and datum paths, tolerances are specified on those vector lengths which are independent of assembly adjustments. That is, tolerances must be specified on those vectors which correspond to manufactured feature dimensions, such as the radius of a corner. Tolerances may also be optionally applied to any angles between the vectors in an assembly which are independent (the angle between two adjacent vectors on the same part) as these represent machined or manufactured angles. Dimensions or angles which correspond to kinematic variables adjust during assembly. Their variations are determined by kinematic analysis [Chase, 1992]. FEATURES Feature variations are variations in shape or form of a part. For example, a machined cylinder will not be perfectly cylindrical, rather, it may be slightly elliptical. The cylinder may still be circular enough to perform its function properly. Feature control tolerances, which include form, location, position and orientation, constrain a variation to fall within a certain tolerance zone. The ANSI Y14.5 standard symbols for form and feature variations are shown in figure 2.4. Current study on feature variations include developing mathematical models for each type of feature variation [Goodrich, 1991; Ward, 1992].

12 Background 11 Flatness Straightness Circularity (roundness) Cylindricity Perpendicularity Angularity Parallelism Profile (of a line) Runout (circular) Concentricity Position Figure 2.4. ANSI Y14.5 Feature Control Symbols. MODEL FILE Once the assembly tolerance model is complete, a neutral file is created containing the information needed for assembly tolerance analysis. This neutral file, used by the analysis software package CATS.BYU, generates a set of algebraic equations to determine the values of the dependent variables. The dependent variables include both the kinematic and assembly variables. Analysis Once vector equations are derived from the geometric model the next step in the analysis process is to linearize the equations for small variations about the nominal by Taylor's series expansion, retaining first order derivatives. Each derivative is evaluated using the nominal dimensions of the dependent and independent variables. The component tolerances dx i are substituted and the system of linearized equations are solved for the corresponding variation in the kinematic variables and resultant assembly dimensions.

13 Background 12 The linearized closed loop equations may be written as [Marler, 1988]: dh x (x i,u j,α k ) = Σ( h x x i dx i) + Σ( h x u j du j) + Σ( h x k dα k) = 0 dh y (x i,u j,α k ) = Σ( h y x i dx i) + Σ( h y u j du j) + Σ( h y k dα k) = 0 dh θ (x i,u j,α k ) = Σ( h θ x i dx i) + Σ( h θ u j du j) + Σ( h θ k dα k) = 0 where dx i are the specified tolerances on the independent dimensions, du j are the resultant variations in the dependent assembly dimensions and dα k are the form and feature variations. In matrix notation these equations become: where [A] {dx} + [B] {du} + [F] {da} = {0} Closed Loop Equations [A] is the matrix of sensitivities to variations in the independent variables x i {dx} is the vector of specified variations of the independent variables dx i [B] is the matrix of sensitivities to variations in the independent variables u j {du} is the vector of unknown variations of the dependent variables du j [F] is the matrix of sensitivities to variations in the form variables α k {da} is the vector of specified variations of the form variables dα k The linearized open loop equations in matrix form may be expressed [Carr, 1992]: {dg} = [C] {dx} + [E] {du} + [G] {da} Open Loop Equations where C, E and G represent sensitivities to variations in dx i, du j and dα k, respectively. dg is the non-zero vector of specified assembly tolerance limits for gaps, rotations or other feature variations. From the above closed and open loop equations, expressions for the predicted accumulation of variations may be generated. Therefore, once the engineering model is created, the equations are derived, dependent variables separated, sensitivities calculated, and expressions for tolerance accumulation computed.

14 Background 13 CATS.BYU allows the designer to use several tolerance analysis options including [Chase, et al. 1987]: Worst case and statistical tolerance accumulation Propagation of variations, including kinematic, dimensional and form variations Percent rejects prediction Accounting for process mean shifts using the Motorola Six Sigma statistical model Tolerance information which has been created on a CAD modeling system may be sent to CATS.BYU for analysis or CATS/PC, the microcomputer implementation of CATS.BYU. CATS estimates tolerance accumulation or stack-up from one of the expressions in Table 2.1. Which one you use depends upon customer requirements, process data available and desired accuracy. Table 2.1. Assembly Tolerance Accumulation Formulas. Worst Case U du= dx i T ASM x i Root Sum Square du= U x i General Root Sum Square 2 dx i 2 T ASM Assures 100% assembly acceptance if all parts are within specification. Costly design model. Requires excessively tight component tolerances Assumes normal distribution and ±3σ tolerances. Some fraction of assemblies will not meet specification. Less costly. tolerances. More versatile. Permits looser component du= Z ASM Six Sigma U x i 2 dx i Z i 2 TASM May adjust Z ASM to obtain desired quality. Most realistic estimates. du= Z ASM U x i 2 dx i 3Cpk i 2 TASM Accounts for process mean shifts and their long-term affects on assembly distribution. In the above table, du is the predicted variation in the resultant assembly sum dimension, dx i is the component dimensional variation, U/ x i is the sensitivity that a variation in dx i has on U, T ASM is the design limit for variations in du. Z ASM and Z i are

15 Background 14 the number of standard deviations corresponding to the assembly and component tolerance limits, respectively, Cpk i is the process capability index and is a measure of the shift in the process mean [Chase et al., 1992]. As presented above, previous modeling techniques included the use of kinematic joints, feature controls and vector loops. Modeling capabilities not fully integrated into the earlier modeling techniques, but developed and implemented through this research include: A comprehensive system for assembly or design specifications Just as feature controls apply to individual parts, assembly tolerance specifications should be applied to assemblies of parts. Manual process variations or fastener clearances These processes are not expressible as constrained kinematic models. Automatic open loop generation Open vector loops are often necessary to represent design constraints on the assembly. Degree of Freedom analysis procedures Only a limited number of redundancy checks are present. A method for detecting degree of freedom redundancy is essential to creating a correct engineering model.

16 CHAPTER 3: DESIGN SPECIFICATIONS Tolerances may be specified on dimensions or features of a production part to establish limits to expected manufacturing variations. Tolerances may also be specified to assemblies of parts. Both tolerances are important for the quality and functionality of an assembly, but differ greatly in meaning and application. Component tolerances are applied to individual part dimensions. The precision specified influences the production processes and tooling selected to make the part. It also affects the choice of inspection methods used to accept or reject the parts as they are produced. Component tolerances are inspected before assembly. Form tolerances are also applied to individual components of a part. Form tolerances are limits on the shape, orientation or location of the features on a part. Feature controls or form tolerances are applied to parts to constrain the variation due to different production methods to acceptable limits. For example, a circle may not be a true circle, but it may be considered circular enough for its purpose. Form tolerances are inspected before assembly to assure quality of the part. Assembly tolerances are applied to the overall or resultant dimensions of an assembly. The resultant dimension of the assembly is generally a function of two or more parts in the assembly. Thus, component tolerances may accumulate or stack up, causing increased variation in the resultant assembly dimensions. Assembly tolerances are inspected after parts are drawn from stock and assembled. Assembly tolerances are applied in order to meet engineering requirements and insure functionality of the final part. Component tolerances are only required when they contribute to assembly variation. Tolerance analysis relates the component tolerances to the assembly tolerances. By this means, the assembly limits determine the allowable variation in the components. Assembly tolerances are an important design tool for engineers. A variety of tolerance specifications are needed in practice. However, there is no accepted standard which covers all design situations. The geometric tolerancing standard, ANSI Y14.5 is a comprehensive system for specifying tolerances, but it applies to individual components or to simple assemblies. This chapter will develop a similar system of design specifications

17 Design Specifications 16 suitable for complex assemblies. The modeling procedures and a short example will also follow. Design Specifications Examination of current engineering practice reveals that assembly tolerances or design specifications are analogous to form tolerances applied to individual components. For every ANSI Y14.5 form tolerance which specifies a datum, there is an assembly tolerance counterpart. Table 3.1 and 3.2 following relates current feature controls applied to components to similar feature controls applied to assembly tolerances. Together, they form a set of assembly tolerance specifications as comprehensive as ANSI Y14.5.

18 Design Specifications 17 Table 3.1. Dimensional tolerance specifications. Component Tolerances Length & Angle Gap Assembly Tolerances u ± du θ ± d θ x ± dx Length u ± du Angle θ ± d θ

19 Design Specifications 18 Table 3.2. Form, orientation and location tolerances specifications. Component Tolerances Assembly Tolerances Parallelism A Parallelism Part C A Part B -A- -A- Perpendicularity & Angularity Perpendicularity & Angularity A A A A -A- Concentricity & Runout Concentricity & Runout -A- A -A- A -A- A A Position -A- x y A B Position -B- A B -B- -A-

20 Design Specifications 19 From the previous tables there are two assembly tolerances or design specifications which are applied to kinematic variables in closed loops, the rest require open loops to be generated. The following table distinguishes the closed and open loop design specifications. Table 3.3. Design specifications according to required loop type. Closed Loop Dependent Length Dependent Angle Open Loop Gap Global Orientation Parallelism and Perpendicularity Relative Angularity Position A detailed description of each of the seven design specifications is presented below. Clearance or Gap A gap specification is the allowed variation in the minimum distance between two parts. It is measured after all parts have been pushed together, leaving the gap as the only clearance remaining. Tolerance limits are set on the maximum and minimum values of the gap. The gap specification, which requires an open loop, is a representation of the ANSI Y14.5 length tolerance. Clearances or interference fits are also assigned tolerances in this manner. For example, in order for a fan to operate properly a certain clearance from the blades must be maintained. Another example is that of a locking computer tape hub (see figure 3.1). The plunger is rigidly set against the base, forcing the arm to slide radially outward. An interference fit between the pad and the reel must be assured (a clearance is shown for clarity). An interference gap specification is assigned between the pad and the reel.

21 Design Specifications 20 C L R L Gap Open Loop R T Plunger Base Arm Reel Pad Figure 3.1. Locking computer tape hub with gap specification and open loop. A tolerance is placed on the interference distance between the two parts. An open loop is then created from the end points of the gap specification. The loop consists of two vectors, R T and R L forming a chain from one side of the gap to the other. The gap will then be the result of the overall dimension R T and the 2-D resultant stack-up R L : Gap = R T - R L The analysis will then determine whether the gap variation will meet the desired specifications. The next two design specifications are assembly tolerances on kinematic variables. No open loops are required since they are already included in a closed vector loop as kinematic variables. The two specifications are dependent length and dependent angle. Dependent Length A dependent length specification constrains the variation of a point's location between two mating parts. It is similar to a length component tolerance. Almost anytime one has a contact with a sliding plane a length specification may be applied. For example, a planar, cylindrical slider or edge slider kinematic joint will contain a sliding plane. A dependent length is a tolerance applied to a kinematic variable in a closed loop, it describes a single kinematic degree of freedom. An example of a critical length would be the vector u 1 in the assembly of figure 3.2. Vectors u 1, u 2, and f are dependent variables. They are therefore, functions of r, a and q. A length specification could be placed on any of the

22 Design Specifications 21 dependent variables. A specification placed on vector u 1 would constrain the vector to the length of u 1 ± du 1 θ u 2 r φ + a r u ± du 1 1 Figure 3.2. Dependent length specification applied to vector u 1. Once analysis is complete, a check is run to see if the kinematic variable is within the desired tolerance range. If the variable is within specification the mechanism should function properly. Dependent Angle A dependent angle specification, comparable to an angle component tolerance, is the angular variation between two mating parts, or it is the dependent angle measured from a cylindrical datum reference frame and a contact point. It describes a single kinematic degree of freedom (see figure 3.3). Ø ± dø Ø ± dø Figure 3.3. Mating part and cylindrical datum relative angle specification. For example, an offset slider crank mechanism shown in figure 3.4. The crank a rotates at a constant speed. The connecting rod b pulls the slider back and forth on the track u. As

23 Design Specifications 22 the crank rotates counterclockwise from position 1 to position 2 the connecting rod pulls the slider from 1 to 2 on the track. This is the forward or power stroke. The return stroke occurs as the crank continues to rotate past position 2 and back to 1. Such reciprocating mechanism are often used in shavers, clippers, and feed mechanisms. A long forward stroke allows smooth transmission of power. The unloaded return stroke is shortened for efficiency. The timing ratio is calculated to be: TR = φ φ A dependent angle specification is applied to f in order to maintain a proper timing ratio. 1 φ a a 2 b b u Figure 3.4. Offset slider crank mechanism. Global Orientation Global orientation is the angular variation between an axis or edge of a part and the global X-axis. This open loop specification is analogous to the ANSI Y14.5 orientation feature tolerance. For example, the remote positioner linkage system may contain a global orientation specification (see figure 3.5). This specification will require the angle of the final link to be within a certain tolerance from the global X-axis.

24 Design Specifications 23 θ input angle 1 X φ 1 φ 2 φ f ± dφf φ 3 Figure 3.5. Remote positioner with global orientation and rotation loop. q 1 represents a specified input angle. The open loop is created using the global coordinates origin as the starting node and the ending linkage as the end point. f 1, f 2, and f 3 represent relative angles between consecutive vectors. f f is the final angle relative to the global X-axis. The rotation loop will be: q 1 + f 1 + f 2 + f 3 + f f = 0 Solving the rotational equation using the worst case and statistical methods result in: df f = dq 1 + df 1 + df 2 + df 3 (WC) dφ f = dθ 1 2 +dφ1 2 +dφ2 2 +dφ3 2 (RSS) df f is then applied to the nominal to see if the variation is within the specification limits. Parallelism and Perpendicularity A parallelism or perpendicularity specification limits the deviation from parallel or perpendicular between two parts, either axes or edges. These specifications are analogous to the parallelism or perpendicularity ANSI Y14.5 for tolerance bands placed on features of parts. Once the two parts are established to be parallel or perpendicular nominally, a tolerance band is placed on the parallelism or perpendicularity of the assembly (see figure 3.6).

25 Design Specifications 24 φ 1 -Aφ 2 φ rel ± dφ rel φ 3 Figure 3.6. Remote positioner with parallelism specification and rotation loop. Since the parts are considered parallel, an open loop will contain one less rotation than if the assembly were modeled as relative angularity. This is deducted from the equation: -f 1 + f 2 + f 3 - f rel = 0 where f rel is considered to nominally be 0 with no input angle. Solving the rotational equation using the worst case and statistical methods result in: df rel = df 1 + df 2 + df 3 (WC) dφrel = dφ 1 2 +dφ2 2 +dφ3 2 (RSS) For the designer's convenience the open loop created will be identical to that of global orientation. Since the open loop is flagged as parallelism, the analysis side will account for the extra node in the loop. The variation is then compared to the parallelism design requirement to see if the linkage functions properly. Relative Angle The relative angle specification is much like the ANSI Y14.5 angularity tolerance only a relative angle specification is the angular variation between two non-mating parts. This design specification allows the designer to correlate an input angle to an output angle of a mechanism. It is therefore a more general case of both global orientation and parallelism or perpendicularity where the nominal input angle is either relative to the ground, 0 or 90 respectively. For example, a linkage may require that the input to output angle ratio be one-to-one. Assigning a relative angle specification between the input linkage and the output linkage will assure that the engineering requirements will be met (see figure 3.7).

26 Design Specifications 25 -Aθ input angle 1 φ 1 φ 2 φ rel ± dφ rel φ 3 Figure 3.7. Remote positioner with angularity specification and rotation loop. As shown above, all angles from the input to the final linkage are required for analysis. The resultant relative angle is determined by summing all the relative angles of the intermediate parts. The open loop will obtain the needed rotation equation: q 1 + f 1 + f 2 + f 3 - f rel = 0 where f rel is considered to nominally be 0 with no input angle. The worst case and statistical variation is then solved for and applied to the relative angle specification: df rel = dq 1 + df 1 + df 2 + df 3 (WC) dφ rel = dθ dφ1 2 + dφ2 2 +dφ3 2 (RSS) Position A position specification places limits on the location of a point in the assembly just as an ANSI Y14.5 position tolerance constrains a point on a part. The point location may vary horizontally and vertically (x, y). By constraining a point on a linkage, for example point P in figure 3.8, the path of the linkage system is guaranteed. Y Y X X dα ± dx P P ± dy a) diameter tolerancing b) rectangular tolerancing Figure 3.8. Linkage system with constrained point in assembly.

27 Design Specifications 26 In order to analyze the position specification an open loop, starting on one part and ending on another, must be specified. The two end points are datums which are each associated with a single part, unlike joints which are associated with two parts. One end point need not be the global reference frame. A tolerance is then placed on the desired point relative to the assembly origin as either a diameter tolerance or rectangular tolerance. In rectangular tolerancing, the x and y coordinates of a point are specified relative to a set of datum axes. Many object to rectangular tolerancing because the permissible position error varies with direction. It is equal to dx in the x direction, dy in the y direction, and dx 2 +dy 2 in the diagonal direction of the rectangular tolerance zone. ANSI Y14.5 has established a non-directional tolerance represented by a circular tolerance zone. No tolerances are applied to the x and y dimensions in this case. The value of the tolerance specifies the diameter of the tolerance zone, da. From the open loop, equations may be generated which will allow the designer to better predict if the assembled mechanism meets the desired specifications. The design specifications modeled for two-dimensions do not include concentricity or runout assembly tolerances. Although runout is easier to measure, perpendicularity may also be used for cylinder faces. By creating a perpendicular specification between the axis of one part and an edge of another part which is perpendicular, the equivalent assembly tolerance is applied (see figure 3.9). A concentricity design specification may be modeled as a gap or clearance in 2-dimensions. C L TOL Figure 3.9. Perpendicularity between cylinder face and center line axis.

28 Design Specifications 27 Summary Assembly tolerances are important design tools for engineers. By incorporating the six design specifications, or assembly tolerances developed into tolerance analysis, a more realistic solution may be provided. This allows for engineers to place constraints on assemblies to insure functionality.

29 Open Loop Generation 28 CHAPTER 4: OPEN LOOP GENERATION The previous chapter presented a new system for specifying limits on assembly variations. By assembly tolerance analysis the designer may predict the affects of variation on design requirements. Open loops, discussed in this chapter, are required to generate the equations necessary to solve for many of these variations. The end product of the assembly tolerance modeling process is a set of vector loops which describe the assembly relationships in terms of geometric and kinematic information (see Chp 2 sec Loops). These loops may be either open or closed loops. Closed loops start and end at the same location and represent kinematic constraints on the assembly. Open loops are used to determine assembly resultants of interest, such as a clearance, orientation or position. They are used to model the engineering constraints on an assembly There must be enough closed loops to solve for all the dependent kinematic variables. There must also be enough open loops to solve for all other assembly resultants of interest. Creating the vector loops is a major portion of the engineering modeling procedure. There are many possible loops in an assembly, but only a few are required for analysis. Not only that, but the loops, whether open or closed, are not unique since each loop may follow a variety of possible paths. These paths may be difficult to define since the choice of loop paths is up to the designer. Below is a brief overview of the existing closed loop theory. Manual Closed Loop Creation When creating vector loops several rules must be followed. By following these rules, enough equations may be generated to solve for dependent kinematic variables and sensitivities. Three beginning modeling rules include: The set of loops must pass through every part and every joint in the assembly.

30 Open Loop Generation 29 No single loop may pass through a part or joint more than once, but it may start and end at the same point. There must be enough loops to solve for all the kinematic variables -- one loop for every three variables. The vector loop passes from mating part to mating part, always crossing through the joints that connect the parts. In a similar fashion, the vector loop enters a part through one joint (using an incoming vector) and leaves through another joint (using an outgoing vector), thus crossing the part. The vector loop always follows paths along associated datum paths created with each joint. While creating vector paths the following three modeling rules apply: A vector loop must pass from one part to the next mating part through a common joint. The path across a part must pass through a Datum Reference Frame (), following the datum path from the incoming joint to the and then following the datum path from the outgoing joint to the, only in reverse. If the path across the part doubles back on itself with an equal and opposite vector, the two vectors cancel each other and will be omitted from the loop. Path 2 Path 1 Figure 4.1. Datum paths define the path across a part.

31 Open Loop Generation 30 Different joints along the vector paths place requirements on the vector loop. These requirements assure the angles and vectors corresponding to the kinematic variables will be present in the assembly, so their variations can be calculated. For a cylindrical slider joint, either the incoming or outgoing vector must be normal to the sliding pane and end at the center of the cylinder. For joints having a sliding plane (planar, cylindrical slider or edge slider), either the incoming or outgoing vector must lie in the sliding plane. For cylindrical joints (parallel cylinders in contact), the path through the joint must start at the center of one cylinder and end at the center of the mating cylinder, passing through the contact point Planar Edge Slider Cylindrical Slider Cylindrical Figure 4.2. Required vector paths through joints. Manual Open Loop Creation A similar theory for manual open loop creation will now be developed. Many of the same closed loop rules apply for open loop generation, but, the differences are important. Open loops describe engineering constraints on our assembly. It is therefore possible to model an entire system without any open loops. This is one major difference between open and closed vector loops. Open loops must pass through at least two parts since design specifications are applied between parts. The parts, whether two or more, contact in some manner, therefore, open loops also require at least one joint. Since open loops are not closed, the starting and ending points must be different to create an open loop. Also, assembly tolerances determine the variation of the entire assembly which contribute to the design specifications. This requires there to be an open loop for each specification. The general rules for open loops are summarized as:

32 Open Loop Generation 31 The loop must pass through at least 2 parts and 1 joint. A single loop may not start and end at the same point. There must be a maximum of one open loop for every design specification. It should be noted that the open loop does not have to pass through every part or every joint. Also, depending on the number of assembly constraints, or assembly tolerances, there may be any number of open loops. Vector paths are created in the same manner as closed loops, the only difference is that the end points of specifications are rectangular datums rather than kinematic joints. Each endpoint is associated with one part and has one datum path. This allows the vector loops to remain open loops. For example, Joint 6 (figure 4.3) requires two datum paths closing the loop from 2 to 5. On the other hand, Specification endpoints 7 and 8 only have one datum path, there is therefore only one vector leaving each endpoint and the loop may not be closed. Datum 1 2 Datum 1 2 Spec Ends 7,8 Joint 3 Joint 3 Joint 6 Datum 4 5 Datum 4 5 Figure 4.3. Open loop specification datum paths versus closed loop datum paths. Open vector loops also have the same joint requirements as closed loops. By automating these procedures, not only will the number of errors be reduced, but also the modeling time. Much of the existing theory for loop automation, the modified Automatic Vector Loop (AVL) algorithm, developed by Larsen and Simmons [Larsen, 1991; Simmons, 1990] has been extended for use in open loop generation. The modified AVL theory will be discussed next. Modified AVL

33 Open Loop Generation 32 The AVL algorithm begins by creating a connectivity matrix. A connectivity or incidence matrix [Larsen, 1991] is a matrix used to map the relationships between parts and joints. By convention, matrix rows represent different parts in the assembly and columns represent individual joints. Each joint touches two parts, so two 1's are placed in each joint column for the two corresponding parts. 0's fill the rest of the column. J4 J5 CYLINDER BLOCK GROUND J7 J9 J11 Cylinder Block Ground J4 J5 J7 J9 J Figure 4.4. Connectivity matrix and corresponding assembly. For example, in figure 4.4, the connectivity matrix indicates that joint 4 connects the Cylinder to the Ground and the Block contacts the Ground at joints 5, 9 and 11. A set of loops is obtained by moving between the 1's in the matrix. A horizontal path between 1's represents crossing a part, while vertical paths between 1's represent transferring between parts through joints. The number of loops required for tolerance analysis [Larsen, 1991] is: Loops = Joints - Parts + 1 For the assembly in figure 4.4: L = = 3 loops VIRTUAL TREE REPRESENTATION Once the connectivity matrix has been constructed, candidate paths are found and compared. An exhaustive search must be made for all possible loop candidates and a selection made of an optimum set, as determined by a set of criteria to be described later. To obtain a candidate closed loop a starting joint and part are chosen from the matrix. A candidate loop is obtained when the tree search completes a loop by returning to its starting joint. A candidate closed loop is shown for the assembly of figure 4.4:

34 Open Loop Generation 33 J4 J5 J7 J9 J11 Cylinder Block Ground Figure 4.5. Closed path representation in a connectivity matrix. To find the next candidate loop, the path is traversed until reaching a place where an alternate path may be explored. The downward search is then continued until the path doubles back on itself, or arrives at the starting joint. This step is iterated until all possible loops have been found [Larsen, 1991]. To keep track of the loops found the 1 elements along the path are changed to -1's. Thus, the -1 paths are visible in the matrix. For example, in figure 4.6, the Block has four joints, indicated by the four ones in its row. The path enters the block through joint 7 (the incoming joint). There are three possible outgoing joints (joints 5, 9 and 11). The first loop exited through joint 9 in figure 4.5. the alternate loop in figure 4.6 exits through joint 5. An additional candidate loop is possible, using joint 11 as the outgoing joint from the Block. J4 J5 J7 J9 J11 Cylinder Block Ground Figure 4.6. Alternate closed path in the connectivity matrix. Candidate loops, once found, are then expanded to include not only the different joints and parts, but also the paths across the parts. This ensures that the vector loop consists of only controlled and kinematic dimensions. The datum paths, or paths from the joint to the of each touching part, are now included in the vector loop and a complete candidate loop is defined [Larsen, 1991].

35 Open Loop Generation 34 The loop found in the connectivity matrix of figure 4.6 can be expanded and constructed as a graphical overlay on the assembly drawings, as shown in figure 4.7. Overlaying this diagram on the assembly reveals the finished vector path through the assembly. Jnt 4 Cylinder Jnt 4 Cylinder Jnt 5 Datum Jnt 7 Jnt 5 Jnt 7 Ground (a) Block Ground Datum Block Figure 4.7. Expanded loop with joints, parts and datums (from figure 4.4) (a) vector loop and (b) overlaid on assembly. (b) A check must be made for redundant dimensions. For example, if the two datum paths overlap, they share common dimensions on the paths back to the. For example, the datum path in figure 4.7 from joint 5 to the Block and back is redundant. These redundant dimensions are not needed for the tolerance analysis and may introduce unrealistic variation into the tolerance model. The redundant dimensions are removed from the candidate loop until a complete candidate loop is defined [Larsen, 1991]. Jnt 4 Cylinder Jnt 7 Jnt 5 Ground Figure 4.8. Completed candidate loop (from figure 4.4). SELECTION OF LOOPS

36 Open Loop Generation 35 A criteria has been established for comparing candidate loops. The first criterion is the number of nodes in the loop. The fewer the nodes, the better the loop. Candidate loops are compared with the current best loop for that starting joint. Thus, if the candidate loop has fewer nodes than the current best loop, the candidate loop becomes the current best loop. The next criterion is physical length. If both loops being compared have the same number of nodes, then the expanded length of the loops are compared. The loop which is shortest dimensionally becomes the current best loop. The current best loop is stored and the process of obtaining candidate loops continues until all possible loop candidates which begin at the selected starting point have been tested. A new starting point for the next closed loop is chosen from a joint which was not used in the first loop. If all the joints have been used in a loop, then the loop will start at a joint which was not a starting joint for any other loop. This searching process continues until the required number of loops is determined using the formula previously stated as [Larsen, 1991].: Loops = Joints - Parts + 1 This is an exhaustive search. The number of possible candidate loops is: n NT = N i -1 where Ni n NT i=1 is the number of joints on part i is the number of parts is the total number of loops possible. For the assembly in figure 4.4 the equation becomes: NT = (2-1)(4-1)(4-1) = 9 loops possible Now that the closed loop AVL theory has been reviewed, a similar development for the AVL open loop theory will be presented. Modified AVL for Open Loops Candidate loops are found and compared much the same as in the modified AVL algorithm. On the other hand, starting and ending points are restricted for each open loop. Open loops must start at one endpoint of a specification and end at the other